On the genus of graphs from commutative rings

Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli 627 012, Tamil Nadu, India Received 11 June 2015; accepted 27 May 2016

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Abstract

LetRbe a commutative ring with non-zero identity. The cozero-divisor graph ofR, denoted byΓ^′(R), is a graph with vertex-set W^∗(R), which is the set of all non-zero non-unit elements ofR, and two distinct verticesxand yinW^∗(R)are adjacent if and only ifx ̸∈ R yandy̸∈ Rx, where forz∈ R,Rzis the ideal generated byz. In this paper, we determine all isomorphism classes of finite commutative ringsR with identity whoseΓ^′(R)has genus one. Also we characterize all non-local rings for which the reduced cozero-divisor graphΓr(R)is planar.

c

⃝2017 Publishing Services by Elsevier B.V. on behalf of Kalasalingam University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords:Genus; Local ring; Nilpotent; Planar graph

1. Introduction

The study of algebraic structures, using the properties of graphs, became an exciting research topic in the past twenty years, leading to many fascinating results and questions. In the literature, there are many papers assigning graphs to groups, semigroups and rings. In 1988, Beck began to investigate the possibility of coloring a commutative ringRby associating to the ring azero-divisor graph, defined as a simple graph, the vertices of which are the elements of the ring R, with two distinct elementsx and ybeing adjacent if and only if x y = 0 [1]. Retaining the original definition, the next decade brought little progress. However, in 1999, Anderson and Livingston [2] modified and studied the zero-divisor graphΓ(R)whose vertices are the nonzero zero-divisors Z(R)^∗ of the commutative ring R. In [3], Khashyarmanesh defined the cozero-divisor graph of commutative rings. The cozero-divisor graph of R, denoted byΓ^′(R), is a graph with vertex-set W^∗(R), which is the set of all non-zero non-unit elements of R, and two distinct verticesx and yinW^∗(R)are adjacent if and only if x ̸∈ R y andy ̸∈ Rx, where forz ∈ R, Rz is the ideal generated byz. In [3,4], some basic results on the structure of this graph were obtained. Also, in [5], the situation whereΓ^′(R)is planar, outer planar and ring graph is investigated. In [6], A. Wilkens et al. defined the reduced cozero-divisor graph of commutative rings. The reduced cozero-divisor graph of R, denoted byΓ_r(R), is a

Peer review under responsibility of Kalasalingam University.

∗Corresponding author.

E-mail addresses:kavithaashmi@gmail.com(S. Kavitha),karthipyi91@yahoo.co.in(R. Kala).

http://dx.doi.org/10.1016/j.akcej.2016.11.006

0972-8600/ c⃝2017 Publishing Services by Elsevier B.V. on behalf of Kalasalingam University. This is an open access article under the CC BY- NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Rxand R yinΩ(R)are adjacent if and only if Rx ̸⊆ R y andR y ̸⊆ Rx. We denote the ring of integers modulon byZn, the field withq elements byFqand the set of all nilpotent elements inRbyN(R). Note thatR^×is the set of all units in R. For any set X, letX^∗denote the nonzero elements of X. For basic definitions on rings, one may refer to [7].

LetS_kdenote the sphere withkhandles, wherekis a nonnegative integer, that is,S_kis an oriented surface of genus k. The genus of a graphG, denotedg(G), is the minimum integernsuch that the graph can be embedded inS_n. For details on the notion of embedding a graph in a surface, see [8]. Intuitively,Gis embedded in a surface if it can be drawn in the surface so that its edges intersect only at their common vertices. A genus 0 graph is called a planar graph and a genus 1 graph is called a toroidal graph. We note here that ifHis a subgraph of a graphG, theng(H)≤g(G). LetGandHbe two graphs with disjoint vertex setsV1,V2and edge setsE1,E2respectively, theirjoinis denoted by G+Hand it consists ofG∪Hand all edges joining every vertex ofV1with every vertex ofV2. For basic definitions on graphs, one may refer [9]. The following results are useful for further reference in this paper.

Example 1.1 ([10,11]).Let(R,m)be a finite local ring Then

|m^∗| R |R|

1 Z4,^Z⟨²x^[x]2⟩ 4 2 Z9,^Z⟨³x^[x]2⟩ 9 3 Z8,^Z⟨²x^[x]3⟩,⟨2x^Z,⁴x^[x]2−2⟩ 8

Z4[x]

⟨2,x⟩²,^Z2[x,y]

⟨x,y⟩² 8

F4[x]

⟨x²⟩, ⟨x^2Z+x+1^4[x] ⟩ 16 4 Z25,^Z⟨⁵x^[x]2⟩ 25

Lemma 1.2. g(Kn)=_{(n−3)(n−4)}

12



if n≥3. In particular, g(Kn)=1if n=5,6,7.

Lemma 1.3. g(K_m,n)= ₍

m−2)(n−2) 4



if m,n ≥ 2. In particular, g(K_4,4) = g(K_3,n) =1 if n = 3,4,5,6. Also g(K5,4)=g(K6,4)=g(Km,3)=2if m=7,8,9,10.

Theorem 1.4. Let R be a finite commutative non-local ring. ThenΓ^′(R)is planar if and only if R is isomorphic to one of the following rings:Z2×Z2×Z2,Z2×F ,Z2×Z4,Z2×^Z2[x]

⟨x²⟩,Z3×F ,Z3×Z4, orZ3×^Z2[x]

⟨x²⟩, where F is a finite field.

Fig. 2.1. A planar embedding ofΓ^′(Z3×Z2×Z2)inS₁.

2. Genus ofΓ^′(R)

In this section, we determine all isomorphism classes of finite commutative non-local rings with identity whose Γ^′(R)has genus one.

Theorem 2.1. Let R=F1× · · · ×Fnbe a finite commutative ring with identity, where each Fj is a field and n≥2.

Then g(Γ^′(R))=1if and only if R is isomorphic to one of the following rings:F4×F4,F4×Z5,Z5×Z5,F4×F7

orZ3×Z2×Z2.

Proof. Suppose g(Γ^′(R)) = 1. Suppose n ≥ 4. Let A = {x1,x2, . . . ,x11}, where x1 = (0,0,0,1,0, . . . ,0), x2 = (0,0,1,0, . . . ,0), x3 = (0,0,1,1,0, . . . ,0), x4 = (0,1,0, . . . ,0), x5 = (1,1,0,0. . . ,0), x6 = (1,0, ,0, . . . , . . . ,0), x7 = (0,1,1,1,0, . . . ,0), x8 = (0,1,1,0, . . . ,0), x9 = (1,0,0,1,0, . . . ,0), x10 = (1,0,1,0, . . . ,0), x11 = (1,0,1,1,0. . . ,0) ∈ W^∗(R). Then the subgraph induced by A inΓ^′(R) contains G^′ (seeFig. 1) as a subgraph and sog(Γ^′(R))≥2. Hencen≤3.

Case 1.n=2.

Note thatΓ^′(R)∼= K_{|F1|−1,|F2|−1}. Sinceg(Γ^′(R))=1, byLemma 1.3,R is isomorphic to one of the following ringsF4×F4,F4×Z5,Z5×Z5orF4×F7.

Case 2.n=3.

Suppose at least two ofFi’s are such that|Fi| ≥ 3. Without loss of generality let|F1| ≥ 3 and|F2| ≥ 3. Let B= {z1, . . . ,z11}, wherez1=(1,0,0),z2=(1,1,0),z3=(1, v1,0),z4=(0,1,1),z5=(0, v1,1),z6=(0,0,1), z7=(0,1,0),z8=(0, v1,0),z9=(1,0,1),z10 =(u1,0,0),z11 =(u1,0,1), 1̸=u1∈F₁^∗, 1̸=v1∈ F₂^∗. Then the subgraph induced byBinΓ^′(R)containsG^′(seeFig. 1) as a subgraph and sog(Γ^′(R))≥2, a contradiction. Hence

|Fi| = |Fj| =2 for somei ̸= j. We assume thatF2=F3=Z2.

Suppose |F₁| ≥ 4. Let B^′ = {b₁, . . . ,b₁₁}, where b₁ = (u₁,0,1),b₂ = (1,0,1),b₃ = (u₂,0,1), b₄ = (u₁,1,0),b₅ = (0,1,0), b₆ = (u₂,1,0), b₇ = (0,0,1),b₈ = (0,1,1),b₉ = (u₂,0,0), b₁₀ = (1,0,0), b₁₁ = (u₁,0,1) ∈ W^∗(R)and 1 ̸= u_i ∈ F₁^∗,i = 1,2. Then the subgraph induced by B^′ inΓ^′(R)contains G^′ (seeFig. 1) as a subgraph and so g(Γ^′(R)) ≥ 2, a contradiction. Hence|F₁| ≤ 3. SinceΓ^′(R)is non-planar, by Theorem 1.4,R Z2×Z2×Z2and henceF₁=Z3.

Converse follows fromLemma 1.3andFig. 2.1.

Remark 2.2. Note that if R = R₁× · · · ×R_n be a commutative ring with identity, where each(R_i,m_i)is a local ring withm_i ̸= {0}andn ≥ 2, then Γ^′(Z4×Z4)is a subgraph ofΓ^′(R). Sinceg(Γ^′(Z4×Z4)) ≥ 2, we have g(Γ^′(R))≥2.

Fig. 2.2. A planar embedding ofΓ^′(Z9×Z2)andΓ^′

 Z3[x]

x² ×Z2

 inS₁.

Fig. 2.3. A planar embedding ofΓ^′(Z8×Z2)inS₁.

Note thatΓ^′(Z8×Z2)∼=Γ^′(R), whereRis isomorphic to one of the following rings:^Z⟨²x^[x]3⟩ ×Z2, ⟨_2x,x^Z4[x]2−2⟩×Z2,

Z4[x]

⟨2,x⟩² ×Z2or ^Z2[x,y]

⟨x,y⟩² ×Z2.

Theorem 2.3. Let R=R₁× · · · ×R_n×F₁× · · · ×F_m be a commutative ring with identity, where each(R_i,m_i)is a local ring withm_i ̸= {0}, F_j is a field and n,m≥ 1. Then g(Γ^′(R))=1if and only if R is isomorphic to one of the following rings:Z4×F4,^Z⟨²x^[x]2⟩ ×F4,Z4×F5,^Z⟨²x^[x]2⟩ ×F5,Z4×F7or ^Z⟨²x^[x]2⟩ ×F7,Z9×Z2,^Z⟨³x^[x]2⟩ ×Z2,Z8×Z2,

Z2[x]

⟨x³⟩ ×Z2, ⟨2x^Z,⁴x^[x]2−2⟩ ×Z2, ^Z4[x]

⟨2,x⟩² ×Z2or ^Z2[x,y]

⟨x,y⟩² ×Z2.

Proof. Assume thatg(Γ^′(R))=1. Supposen+m ≥3. LetΩ = {x₁, . . . ,x₁₁}, wherex₁ =(1,0,0, . . . ,0),x₂= (u,0,0, . . . ,0),x₃ =(a,0,0, . . . ,0),x₄=(0,0,1,0, . . . ,0),x₅ =(0,1,0, . . . ,0),x₆=(0,1,1,0, . . . ,0),x₇ = (1,1,0, . . . ,0),x8 =(u,1,0, . . . ,0),x9=(1,0,1,0, . . . ,0),x10 =(u,0,1,0, . . . ,0),b11 =(a,0,1,0, . . . ,0)∈ W^∗(R), 1̸=u ∈ R₁^×anda ∈m^∗₁. Then the subgraph induced byΩ inΓ^′(R)containsG(seeFig. 1) as a subgraph and sog(Γ^′(R))≥2, a contradiction. HenceR=R1×F1.

Suppose|F1| ≥ 3. If|m^∗₁| ≥ 2, then|R^×₁| ≥ 4. LetΩ₁ = {(a,0), (b,0), (u1,0), (u2,0), (u3,0), (u4,0), (0,1), (0, v1), (a,1), (a, v1), (b,1), (b, v1) : a,b ∈ m^∗₁, u_i ∈ R^×₁, and 1 ̸= v1 ∈ F₁^∗} ⊆ W^∗(R). Then the subgraph induced by Ω₁ contains K_6,6 as a subgraph of Γ^′(R) and so g(Γ^′(R)) ≥ g(K_6,6) ≥ 2, a contradiction. Hence

|m^∗₁| =1 and soR₁∼=Z4or ^Z⟨²x^[x]2⟩. ByTheorem 1.4,F₁̸=Z3and|F₁| ≥4.

If|F₁| ≥8, thenK_3,7is a subgraph ofΓ^′(R)and byLemma 1.3,g(Γ^′(R))≥2, a contradiction. Hence|F₁| ≤7 and soF₁is isomorphic to one of the following fields:F4,F5orF7.

Fig. 2.4. A planar embedding ofΓ(Z4×F4)andΓ^′

 Z2[x]

x² ×F4

 .

Fig. 2.5. A planar embedding ofΓ(Z4×F5)andΓ^′

 Z2[x]

x² ×F5

 .

Suppose|m^∗₁| ≥ 2. Then by above argument, we have F1 = Z2. If|m^∗₁| ≥ 4, then|R₁^×| ≥ 5 and so K5,5 is a subgraph ofΓ^′(R). ByLemma 1.3,g(Γ^′(R))≥2, a contradiction. Hence|m^∗₁| =2 or 3 and soR1is isomorphic to one of the following rings:

Z9,Z3[x]

x² ,Z8,Z2[x]

x³ , Z4[x]

2x,x²−2, Z4[x]

⟨2,x⟩²,Z2[x,y]

⟨x,y⟩² ,F4[x]

x² or Z4[x]

x²+x+1. IfR₁∼=^F4[x]

⟨x²⟩ or ⟨x^2Z+x^4[x]+1⟩, then|R₁^×| =12 and soK_4,12is a subgraph ofΓ^′(R). ByLemma 1.3,g(Γ^′(R))≥2, a contradiction. HenceR1 ^F⟨⁴x^[x]2⟩ and⟨x^2Z+x+1^4[x] ⟩.

Converse follows fromFigs. 2.2–2.6.

3. Planarity ofΓ_r(R)

LetRbe a commutative ring with identity. Thereduced cozero-divisor graphof R, denoted byΓ_r(R), is a graph with vertex-setΩ(R), which is the set of all non-zero nontrivial principal ideals of R, and two distinct vertices Rx andR yinΩ(R)are adjacent if and only ifRx ̸⊆ R y andR y ̸⊆ Rx. In this section, we determine all isomorphism classes of finite commutative non-local rings with identity whoseΓ_r(R)is planar.

Theorem 3.1. Let R=F1× · · · ×Fnbe a finite commutative ring with identity, where each Fj is a field and n≥2.

ThenΓ_r(R)is planar if and only if R is isomorphic to one of the following rings: F1×F2×F3or F1×F2. Proof. SupposeR∼=F₁×F₂×F₃,F₁×F₂. Then it is easy to see thatΓ_r(R)is planar (see,Fig. 3.1).

Fig. 2.6. A planar embedding ofΓ^′(Z4×F7)andΓ^′

 Z2[x]

x² ×F7

 .

Fig. 3.1. Planar Embedding of Reduced Co-zero divisor graph.

AssumeΓ_r(R)is planar. Supposen ≥ 4. LetA = {x₁,x₂,x₃,x₄,x₅,x₆} ⊆ V(Γ_r(R)), wherex₁ = F₁×(0)× (0)×(0)× · · · ×(0),x₂= F₁×(0)×(0)×F₄×(0)× · · · ×(0),x₃ = F₁×F₂×(0)×(0)× · · · ×(0),x₄ = 0×(0)×F₃×F₄×(0)× · · · ×(0),x₅=(0)×F₂×F₃×F₄×(0)× · · · ×(0),x₆=(0)×F₂×F₃×(0)×(0)× · · · ×(0). Then the subgraph induced by A in Γ_r(R) contains K_3,3 as a subgraph, a contradiction. Hence n ≤ 3 and so

R∼=F₁×F₂×F₃or F₁×F₂.

Theorem 3.2. Let R = R1×R2× · · · ×Rn be a commutative ring with identity1, where each(Ri,m_i)is a local ring withm_i ̸= {0}and n ≥2. ThenΓ_r(R)is planar if and only if R∼= R1×R2such thatm_i is the only non-zero principal ideal in R_i.

Proof. SupposeR = R₁×R₂such thatm_i is the only non-zero principal ideal inR_i fori =1,2. Then it is easy to see that the graphΓ_r(R)is planar (seeFig. 3.2).

Assume that Γ_r(R) is planar. Suppose n ≥ 3. Let A = {x1,x2,x3,x4,x5,x6} ⊆ V(Γ_r(R)), where x1 = (0)×R₂×(0)× · · · ×(0), x₂ = (0)× R₂× I ×(0)× · · · ×(0), x₃ = (0)×R₂× R₃×(0)× · · · ×(0), x₄ = R₁×(0)× · · · ×(0),x₅ = R₁×(0)×I ×(0)× · · · ×(0),x₆ = R₁×(0)×R₃×(0)× · · · × · · · ×(0). Here I is non-zero principal ideal in R₃. Then the subgraph induced by AinΓ_r(R)containsK_3,3as a subgraph, a contradiction. Hencen=2.

SupposeRi contains at least two non-zero principal ideals for somei. Without loss of generality, we assume that i=2. LetI1= ⟨x⟩,I2= ⟨y⟩be two distinct principal ideals inR2.

Case 1.I₁̸⊆I₂andI₂̸⊆I₁. LetB= {x₁,x₂, . . . ,x₆} ⊆V(Γ_r(R)), wherex₁=(0)×I₁,x₂=R₁×I₁,x₃=J₁×R₂, x₄=(0)×I₂,x₅=R₁×I₂,x₆=J₁×I₂andJ₁is a principal ideal inR₁. Then the subgraph induced byBinΓ_r(R) containsK_3,3as a subgraph, a contradiction.

Case 2. I1 ⊂ I2or I2 ⊂ I1. Without loss of generality, we assume thatI1 ⊂ I2. LetC = {y1,y2, . . . ,y7,y8} ⊆ V(Γ_r(R)), where y1 = R1×(0),y2 = R1×I2, y3 = (0)× R2, y4 = J1×I2, y5 = (0)×I2, y6 = J1× R2, y₇= J₁×(0),y₈= R₁×I₂andJ₁is a principal ideal in R₁. Then the subgraph induced byCinΓ_r(R)contains a subdivision ofK_3,3, a contradiction. HenceR_i contains only one principal ideal. We claim that this principal ideal is maximal. It is enough to prove thatm₁is a principal ideal in R₁.

Supposem₁is not a principal ideal inR1. Then there exists two distinct elementsx,y∈m₁such that⟨x⟩and⟨y⟩ are distinct ideals inR1. ThusR1contains two non-zero principal ideals and by Cases 1 and 2,Γ_r(R)containsK3,3

as a subgraph, a contradiction. Hencem_i is principal for alli=1,2.

Converse follows fromFig. 3.2.

Theorem 3.3. Let R=R₁×· · ·×R_n×F₁×· · ·×F_mbe a finite commutative ring with identity, where each(R_i,m_i)is a local ring and each F_j is a field, m,n≥1. ThenΓ_r(R)is planar if and only if R satisfies the following conditions:

(i)n+m=2

(ii)There exists only two non-zero principal ideals⟨a₁⟩,⟨a₂⟩in R₁such that⟨a₁⟩ ̸⊆ ⟨a₂⟩and⟨a₂⟩ ̸⊆ ⟨a₁⟩. (iii)m₁= ⟨a⟩is a principal ideal with nilpotency at most k=4and

if k=2, then⟨a⟩is the only ideal in R₁ if k=3, then⟨a⟩,

a²

are the only ideals in R₁ if k=3, then⟨a⟩,

a² ,

a³

are the only ideals in R₁.

Proof. SupposeRis isomorphic to R₁×F₁whereR₁has at most three proper ideals. It is easy to see thatΓ_r(R)is planar (seeFigs. 3.3and3.4).

Conversely, assume thatΓ_r(R)is planar. Suppose n ≥ 2. Let A = {x₁,x₂,x₃,y₁,y₂,y₃} ⊆ V(Γ_r(R)) where x₁ = ⟨x⟩ × ⟨y⟩ × · · · × R_n × F₁ ×(0)× · · · ×(0), x₂ = (0)× R₂ × · · · × R_n × F₁ ×(0)× · · · × (0), x₃= ⟨0⟩ × ⟨y⟩ ×R₃× · · · ×R_n×F₁×(0)× · · · ×(0),y₁=R₁×(0)× · · · ×(0),y₂=R₁×R₂×(0)× · · · ×(0), y₃ = R₁×(0)× · · · ×(0)×F₁×(0)× · · · ×(0). Then the subgraph induced by A inΓ_r(R)containsK_3,3as a subgraph, a contradiction. Hencen=1.

Supposem≥2. Thenn+m≥3. LetB= {z₁,z₂,z₃,u₁,u₂,u₃} ⊆V(Γ_r(R)), wherez₁= ⟨x⟩ ×F₁×F₂×(0)×

· · · ×(0),z₂=(0)×F₁×F₂×(0)× · · · ×(0),x₃=(0)×(0)×F₂×(0)× · · · ×(0),u₁=R₁×F₁×(0)× · · · ×(0), u₂=R₁×F₁×(0)× · · · ×(0),u₃=R₁×(0)× · · · ×(0). Then the subgraph induced byBinΓ_r(R)containsK_3,3 as a subgraph, a contradiction. Hencem=1 and soR =R₁×F₁.

SinceRis finite,R₁is an Artinian ring and so every ideal inR₁is finitely generated. LetΩ = {a₁, . . . ,a_t :a_i ∈ R₁,a_i ̸=a_j,fori ̸= j}be a minimal generating set form₁inR₁. Thent ≥1 and⟨a_i⟩ ̸⊆

a_j

for alli̸= j.

Supposet ≥ 3. Let B = {x₁,x₂,x₃,x₄,x₅,x₆} wherex₁ = ⟨a₁⟩ ×(0), x₂ = ⟨a₂⟩ ×(0), x₃ = R₁×(0), x₄ = ⟨a₃⟩ ×(0),x₅ = ⟨a₃⟩ ×F₁,x₆ = (0)×F₁. Then the subgraph induced by B inΓ_R(R)contains K_3,3as a subgraph, a contradiction. Hencet≤2.

Fig. 3.4. Γ_r(R₁×F₁)andt=1.

Assume thatt =2. Suppose⟨b1⟩is a non-zero ideal inR1such that⟨b1⟩ ⊂ ⟨a1⟩. Then the subgraph induced by {⟨a1⟩ ×(0),⟨b1⟩ ×(0),R1×(0),⟨a2⟩ ×(0),⟨a2⟩ ×F1, (0)×F1}inΓ_r(R)containsK3,3as a subgraph, a contradiction.

Hence⟨a1⟩,⟨a2⟩are the only non-zero principal ideals inR1.

Assume that t = 1. Thenm₁ is a principal ideal generated by a1. Since R1 is Artinian, m₁ is nil-ideal with nilpotencykand som^k₁=(0),m^k−1₁ ̸=(0). From this, we get(0)⊂

 a₁^k−1

⊂

 a₁^k−2

⊂ · · · ⊂ ⟨a₁⟩are the only ideals in R₁. Supposek ≥ 5. Then the subgraph induced by{⟨a₁⟩ ×(0),⟨0⟩ ×F₁,R₁×(0),

a²₁

×F₁, a³₁

×F₁, a₁⁴

× F₁,

a₁²×(0),

a₁³×(0)}inΓ_r(R)contains a subdivision ofK_3,3, a contradiction. Hencek≤4 and so R₁contains at most three non-zero principal ideals.

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